An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular cases, the minimum in the problem is known to be equal to the tropical spectral radius of the matrix. We examine the problem in the common ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 2014]]>The homogeneous system of matrix equations (XTA+AX,X TB+BX)=(0,0), where (A,B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A,B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 438, no. 8, 2013]]>The basic objects in this paper are monotonically nondecreasing n×n matrix functions D(·) defined on some open interval ı=(a,b) of R and their limit values D(a) and D(b) at the endpoints a and b which are, in general, selfadjoint relations in Cn. Certain space decompositions induced by the matrix function D(·) are made explicit by means ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 2012]]>The minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 436, no. 12, 2012]]>Computing the singular values and vectors of a matrix is a crucial kernel in numerous scientific and industrial applications. As such, numerous methods have been proposed to handle this problem in a computationally efficient way. This paper considers a family of methods for incrementally computing the dominant SVD of a large matrix A. Specifically, we describe a unification of a ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 436, no. 8, pp. 2866-2888, 2012]]>Straightforward solution of discrete ill-posed least-squares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 436, no. 10, 2012]]>V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a miniversal deformation of matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct a miniversal deformation ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 436, no. 7, 2012]]>The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by Cn the cycle, and Pn6 the unicyclic graph obtained by connecting a vertex of C6 with a leaf of Pn-6. Caporossi et al. conjectured that the unicyclic graph with maximal ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 5, pp. 1370-1377, 2011]]>Let V be a linear subspace of Mn,p(K) with codimension lesser than n, where K is an arbitrary field and n⩾p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K≃F2. Here, we give a sufficient condition on codim V for V ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 1, pp. 147-151, 2011]]>Let V be a vector space of dimension n over any field F. Extreme values for the possible dimension of a linear subspace of EndF(V) with a particular property are considered in two specific cases. It is shown that if E1 is a subspace of EndF(V) and there exists an endomorphism g of V, not in E1, such ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1580-1587, 2011]]>The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Cn denote the cycle of order n and Pn6,6 the graph obtained from joining two cycles C6 by a path Pn-12 with its two leaves. Let Bn denote the class ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 4, pp. 804-810, 2011]]>Let R be a commutative ring with identity, A and B be unital algebras over R and M be a unital (it A,it B)-bimodule. Let T=AM0B be the triangular algebra consisting of A, it Band M. Motivated by the work of Cheung [14] we mainly consider the question whether every higher derivation on a triangular algebra is ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 5, pp. 1034-1054, 2011]]>We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1657-1665, 2011]]>A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 247-258, 2011]]>We characterize the ACI-matrices all of whose completions have the same rank, determine the largest number of indeterminates in such partial matrices of a given size, and determine the partial matrices that attain this largest number.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 8, pp. 1956-1967, 2011]]>Let G be a graph on n vertices, and let λ1,λ2,…,λn be its eigenvalues. The Estrada index is defined as EE(G)=∑i=1neλi. We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 215-223, 2011]]>In this work, we consider the so-called Lur’e matrix equations that arise e.g. in model reduction and linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 152-173, 2011]]>Classical information geometry has emerged from the study of geometrical aspect of the statistical estimation. Cencov characterized the Fisher metric as a canonical metric on probability simplexes Sn-1={(x1,⋯,xn)∈R+n:∑xi=1} from a statistical point of view, and Campbell extended the characterization of the Fisher metric from probability simplexes to positive cone R+n. In quantum information ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 224-231, 2011]]>The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 1, pp. 1-59, 2011]]>We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a {non-Hermitian} operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 327-342, 2011]]>Let Gnr be the class of all connected graphs of order n with r pendent vertices. In this paper, we determine the unique graph with minimal distance spectral radius in Gnr. In addition, we determine the unique graph with maximal distance spectral radius in Gnr for each r∈{2,3,n-3,n-2,n-1}.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 11, pp. 2828-2836, 2011]]>We give in this paper a group of closed-form formulas for the maximal and minimal ranks and inertias of the linear Hermitian matrix function A-BX-(BX)* with respect to a variable matrix X. As applications, we derive the extremal ranks and inertias of the matrices X±X*, where X is a solution to the matrix equation AXB=C, and ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 10, pp. 2109-2139, 2011]]>In this paper, we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in C∗-algebras, extending rank conditions for matrices and range conditions for Hilbert space operators.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 5, pp. 1388-1394, 2011]]>udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 3, pp. 687-716, 2011]]>Denote by T(n,q,w1,w2,…,wn-1) the set of n-vertex weighted trees with matching number q and fixed positive weight set Wn-1={w1,w2,…,wn-1}, where w1⩾w2⩾⋯⩾wn-1>0. Tan [S.W. Tan, On the sharp upper bound of spectral radius of weighted trees, J. Math. Res. Exposition 29 (2009) 293–...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 6, pp. 1202-1212, 2011]]>In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 1107-1118, 2011]]>In this paper we prove two consequences of the subnormal character of the Hessenberg matrix D when the hermitian matrix M of an inner product is a moment matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 9, pp. 2314-2320, 2011]]>Solution of homogeneous linear systems of equations is a basic operation of matrix computations. The customary algorithms rely on pivoting, orthogonalization and SVD, but we employ randomized preprocessing instead. This enables us to accelerate the solution dramatically, both in terms of the estimated arithmetic cost and the observed CPU time. The approach is effective in the cases of both general ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 854-879, 2011]]>Let A be an n×n complex matrix and c=(c1,c2,…,cn) a real n-tuple. The c-numerical range of A is defined as the setWc(A)=∑j=1ncjxj∗Axj:{x1,x2,…,xn}isanorthonormalbasisforCn.When c=(1,0,…,0), Wc(A) becomes the classical numerical range of A which is often defined as the setW(A)={x∗Ax:...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 615-624, 2011]]>We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin–Staszewski relative ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 10, pp. 2158-2169, 2011]]>In this paper, we investigate the ordering on a semiring of monotone doubly stochastic transition matrices in Shorrocks’ sense. We identify a class of an equilibrium index of mobility that induces the full ordering in a semiring, while this ordering is compatible with Dardanoni’s partial ordering on a set of monotone primitive irreducible doubly stochastic matrices.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1585-1597, 2011]]>Let G be a simple graph with vertices v1,v2,…,vn, of degrees Δ=d1⩾d2⩾⋯⩾dn=δ, respectively. Let A be the (0,1)-adjacency matrix of G and D be the diagonal matrix diag(d1,d2,…,dn). Q(G)=D+A is called the signless Laplacian of G. The largest eigenvalue of Q(G) is called the signless Laplacian ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 8, pp. 1813-1822, 2011]]>In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 625-635, 2011]]>The inverse mean first passage time problem is given a positive matrix M∈Rn,n, then when does there exist an n-state discrete-time homogeneous ergodic Markov chain C, whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 7, pp. 1620-1630, 2011]]>Let A be a unital prime algebra with a nontrivial idempotent over a field F. For any scalar ξ∈F, all additive maps L:A→A satisfying [L(A),B]ξ+[A,L(B)]ξ=0 whenever [A,B]ξ=0 are characterized and the relation of L to the derivations are revealed, where [A,B]ξ=AB-ξBA is the ξ-...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 669-682, 2011]]>The companion factorization for nonsingular matrices belonging to the general linear group GL(n;C) is studied here. The entries in the last row of the companion matrices are explicitly represented in terms of determinants of proper sub-matrices of the matrix being factorized. The computation of the inverse of a nonsingular matrix is given as an application of this ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 5, pp. 1261-1271, 2011]]>Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfyingδ([x,y])=[δ(x),y]+[x,δ(y)]for any x,y∈T with xy=0 (resp. xy=p, where p is the standard idempotent of T), then ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 5, pp. 1137-1146, 2011]]>The performance of multi-agent systems is an important issue. In this paper, it is focused on consensus speed for multi-agent systems with double-integrator dynamics and fixed undirected graphes under a kind of consensus protocols. It is revealed that, under some conditions, the maximum consensus speed is determined by the largest and the smallest nonzero eigenvalues of the ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 294-306, 2011]]>The main goal of this paper is the study of an analogue of the matricial Schur problem for the Potapov class PJ(D) of J-contractive meromorphic functions in the open unit disk D. Using an approach which is based on an analysis of J-central J-Potapov functions, we will solve this problem in the most general case. Moreover, ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 741-784, 2011]]>In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 4, pp. 778-803, 2011]]>Several decompositions of the orthogonal projector PX=X(X′X)−X′ are proposed with a prospect of their use in constrained principal component analysis (CPCA). In CPCA, the main data matrix X is first decomposed into several additive components by the row side and/or column side predictor variables G and H. The decomposed components are then subjected to singular value ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 12, pp. 2539-2555, 2011]]>For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 990-999, 2011]]>Given a pair of distinct eigenvalues (λ1,λ2) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form Qd(λ)00q(λ) having the same eigenvalue s as Q(λ), with Qd(λ) an (n-1)×(n-1) quadratic matrix ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 3, pp. 464-479, 2011]]>We present necessary and sufficient conditions for an n×n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n×n complex matrix A.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 2, pp. 409-419, 2011]]>We show that if A is an n-by-n (n⩾3) matrix of the form0a10⋱⋱an-1an0,then the boundary of its numerical range contains a line segment if and only if the aj’s are nonzero and the numerical ranges of the (n-1)-by-(n-1) principal submatrices of A are all equal. For n=3, this is the case ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 2, pp. 243-254, 2011]]>We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona G∘H of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 5, pp. 998-1007, 2011]]>Let A be a Banach algebra with unity I and M be a unital Banach A-bimodule. A family of continuous additive mappings D=(δi)i∈N from A into M is called a higher derivable mapping at X, if δn(AB)=∑i+j=nδi(A)δj(B) for any A,B∈A with AB=X. In this paper, we show that D ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 2, pp. 463-474, 2011]]>This paper is an exposition of W.B. Arveson’s complete invariant for the unitary similarity of complex, irreducible matrices.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 4, pp. 769-777, 2011]]>In this paper, we compute the natural density of the set of k×n integer matrices that can be extended to an invertible n×n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form Ok×(n-k)Ik. Connections with Cesàro’s Theorem on the density of coprime integers and Quillen–Suslin’s Theorem are...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 5, pp. 1319-1324, 2011]]>Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 89-103, 2011]]>For a graph G=(V,E) with V={1,…,n}, let S(G) be the set of all real symmetric n×n matrices A=[ai,j] with ai,j≠0, i≠j if and only if ij∈E. We prove the following results. If G is the complement of a partial k-tree H, then there exists a positive semidefinite matrix A∈S(G) ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1468-1474, 2011]]>In this paper, self-adjoint extensions for second-order symmetric linear difference equations with real coefficients are studied. By applying the Glazman–Krein–Naimark theory for Hermitian subspaces, both self-adjoint subspace extensions and self-adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions, where the two endpoints may be regular or singular.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 903-930, 2011]]>For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix.For Δ⩾3 and t⩾3, denote by Ta(Δ,t) (or simply Ta) the tree formed from a path Pt on t vertices by attaching Δ-1P2’s on each end of the path Pt, and Tb(...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 9, pp. 2272-2284, 2011]]>We study various aspects of how certain positivity assumptions on complex matrix semigroups affect their structure. Our main result is that every irreducible group of complex matrices with nonnegative diagonal entries is simultaneously similar to a group of weighted permutations. We also consider the corresponding question for semigroups and discuss the effect of the assumption that a fixed linear functional ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 801-812, 2011]]>Let (A,B)∈Cn×n×Cn×m and M be an (A,B)-invariant subspace. In this paper the following results are presented: (i) If M∩ImB={0}, necessary and sufficient conditions for the Lipschitz stability of M are given. (ii) If M contains the controllability subspace of the pair (A,B), sufficient conditions for the Lipschitz stability of the subspace M are given.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 1137-1162, 2011]]>We study the eigenvalues of matrix problems involving Jacobi and cyclic Jacobi matrices as functions of certain entries. Of particular interest are the limits of the eigenvalues as these entries approach infinity. Our approach is to use the recently discovered equivalence between these problems and a class of Sturm–Liouville problems and then to apply the Sturm–Liouville theory.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 7, pp. 1648-1655, 2011]]>The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let T(n,γ) be the set of trees of order n and with domination number γ. In this paper, we characterize the tree from T(n,γ) with the minimal energy, and determine the tree from T(n,...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2382-2393, 2011]]>Let L(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from L(X) onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from L(X) onto itself that preserve the local spectral radius at a ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 6, pp. 1473-1478, 2011]]>The energy of an (edge)-weighted graph is the sum of the absolute values of the eigenvalues of its (weighted) adjacency matrix. We study how the energy of a weighted graph changes when the weights change. We give some sufficient conditions so that the energy of a weighted graph increases when the positive weight increases. We also characterize some classes ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2425-2431, 2011]]>Described is a not-a-priori-exponential algorithm which for each n×n interval matrix A and for each interval n-vector in a finite number of steps either computes the interval hull of the solution set of the system of interval linear equations Ax=b, or finds a singular matrix S∈A.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 2, pp. 193-201, 2011]]>Let T be a continuous map of the space of complex n×n matrices into itself satisfying T(0)=0 such that the spectrum of T(x)-T(y) is always a subset of the spectrum of x-y. There exists then an invertible n×n matrix u such that either T(a)=uau-1 for all a or T(a)=uatu-...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 11, pp. 2674-2680, 2011]]>We study the algebraic structure of the semigroup of all 2×2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1612-1625, 2011]]>We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C*-algebras and the unitarily invariant norms on matrices are presented.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 8, pp. 1981-1987, 2011]]>Let R be a 2-torsion free commutative ring with identity, A,B be unital algebras over R and M be a unital (A,B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let T=AM0B be the triangular algebra consisting of A,B and M, and let d be an R-...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 259-284, 2011]]>Associated with an n×n matrix polynomial of degree ℓ,P(λ)=∑j=0ℓλjAj, are the eigenvalue problem P(λ)x=0 and the linear system problem P(ω)x=b, where in the latter case x is to be computed for many values of the parameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 3, pp. 623-640, 2011]]>For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)⩽-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 12, pp. 2404-2412, 2011]]>In this paper, we shall give a survey of applications of the theory of graph spectra to Computer Science. Eigenvalues and eigenvectors of several graph matrices appear in numerous papers on various subjects relevant to information and communication technologies. In particular, we survey applications in modeling and searching Internet, in computer vision, data mining, multiprocessor systems, statistical databases, and in ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1545-1562, 2011]]>In an earlier paper (R. Bhatia, T. Jain, Higher order derivatives and perturbation bounds for determinants, Linear Algebra Appl. 431 (2009) 2102–2108) we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant. In this paper we continue that work, and find expressions for the derivatives of all orders for the antisymmetric ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 5, pp. 1111-1121, 2011]]>We provide positive answers to some open questions presented recently by Kim and Shader on a continuity-like property of the P-vertices of nonsingular matrices whose graph is a path. A criterion for matrices associated with more general trees to have at most n−1 P-vertices is established. The cases of the cycles and stars are also analyzed. Several ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 2, pp. 514-525, 2011]]>We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 9, pp. 2132-2139, 2011]]>Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1563-1579, 2011]]>We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Gröbner bases of modules in F[x]q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 104-116, 2011]]>The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as LEE(G)=∑i=1neμi, where μ1,μ2,…,μn are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 8, pp. 2065-2076, 2011]]>Let An,n∈N, be a sequence of k×k matrices which converge to a matrix A as n→∞. It is shown that if xn,n∈N, is a sequence of nonnegative nonzero vectors such thatxn+1=Anxn,n∈N, then ρ=limn→∞‖xn‖n is an eigenvalue of the limiting matrix A with a nonnegative eigenvector. This result implies the weak form of the Perron–Frobenius ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 2, pp. 490-500, 2011]]>The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2462-2467, 2011]]>Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless Laplacian.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2570-2584, 2011]]>In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended p-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs. ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2432-2450, 2011]]>We motivate computations in a multifunctional networked system as instances of algebraic path problems on labeled graphs. We illustrate, using examples, that composition operators used in many function computations in a networked system follow semiring axioms. We present an abstract framework, using a special idempotent semiring algebraic path problem, to handle multiple metrics for composition. We show that using different ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1494-1512, 2011]]>Classical algebraic multigrid theory relies on the fact that the system matrix is positive definite. We extend this theory to cover the positive semidefinite case as well, by formulating semiconvergence results for these singular systems. For the class of irreducible diagonal dominant singular M-matrices we show that the requirements of the developed theory hold and that the coarse level ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 11, pp. 2225-2243, 2011]]>Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[ai,j] with ai,j≠0, i≠j if and only if ij is an edge of G. Denote by I↗(G) the set of all pairs (p,q) of natural numbers such that there exists ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 10, pp. 2197-2203, 2011]]>For a pair of n×n Hermitian matrices H and K, a real ternary homogeneous polynomial defined by F(t,x,y)=det(tIn+xH+yK) is hyperbolic with respect to (1,0,0). The Fiedler conjecture (or Lax conjecture) is recently affirmed, namely, for any real ternary hyperbolic polynomial F(t,x,y), there exist real symmetric matrices S1 and S2 such ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 6, pp. 1277-1284, 2011]]>Let Mn(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from Mn(R)×Mn(R) to V satisfies the condition that {u,u}={e,u} whenever u2=u, then there exists a linear map ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 11, pp. 2889-2895, 2011]]>Let G be a graph of order n and μ(G,λ)=∑k=0n(-1)kckλn-k the Laplacian characteristic polynomial of G. Zhou and Gutman [19] proved that among all trees of order n, the kth coefficient ck is largest when the tree is a path and is smallest for a star. In this paper, for two given positive ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 1, pp. 152-162, 2011]]>In this article, a classical result of Cottle and Veinott characterizing least elements of polyhedral sets in terms of nonnegative left-inverses is extended to characterize nonnegativity of some of the most important generalized inverses. We also present generalizations of the other results of these authors for the case of the Moore–Penrose inverse.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 12, pp. 2448-2455, 2011]]>Let G be a simple graph with vertices v1,v2,⋯,vn, of degrees Δ=d1⩾d2⩾⋯⩾dn=δ, respectively. Let A be the (0,1)-adjacency matrix of G and D be the diagonal matrix diag(d1,d2,⋯,dn). Q(G)=D+A is called the signless Laplacian of G. In this paper, we give some sharp bounds on signless Laplacian ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 3, pp. 683-687, 2011]]>We characterize the sets X of all products PQ, and Y of all products PQP, where P,Q run over all orthogonal projections and we solve the problems argmin{‖P-Q‖:(P,Q)∈Z}, for Z=X or Y. We also determine the polar decompositions and Moore–Penrose pseudoinverses of elements of X.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1594-1609, 2011]]>The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 11, pp. 2639-2657, 2011]]>Given a sequence {An} of matrices An of increasing dimension dn with dk>dq for k>q, k,q∈N, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 4, pp. 1163-1170, 2011]]>We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these objects inside the tropical Grassmannian of planes in projective 5-space, and we explore a small tropical Hilbert scheme.

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1778-1785, 2011]]>Datta et al. solved the partial pole placement problem for the symmetric definite quadratic eigenvalue problem where part of the spectrum is relocated to predetermined locations and the rest of the spectrum remains unchanged. In this paper, the problem is solved by a hybrid combination of this result and the method of receptances. This allows for the partial assignment of ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 7, pp. 1689-1696, 2011]]>The rank-sum, rank-product, and rank-union inequalities for Gondran–Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1769-1777, 2011]]>Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 2, pp. 559-581, 2011]]>Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In [5], Cvetković et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q2) and the ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2420-2424, 2011]]>Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is the set σπ(A)={z∈σ(A):|z|=maxω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 6, pp. 1326-1335, 2011]]>In 1970s, Gutman introduced the concept of the energy E(G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 10, pp. 2334-2346, 2011]]>Let A be a unital algebra and let M be a unitary A-bimodule. We consider generalized Lie derivations mapping from A to M. Our results are applied to triangular algebras, in particular to nest algebras and (block) upper triangular matrix algebras. We prove that under certain conditions each generalized Lie derivation of a triangular algebra A is the sum ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 6, pp. 1532-1544, 2011]]>describe how to find the general solution of the matrix equation XA+AXT=0, with A∈Cn×n, which allows us to determine the dimension of its solution space. This result has immediate applications in the theory of congruence orbits of matrices in Cn×n, because the set {XA+AXT:X∈Cn×n} is the tangent space at A to the congruence orbit of A. Hence, the ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 1, pp. 44-67, 2011]]>We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 7, pp. 1649-1656, 2011]]>This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)m=(QP)m and (PQ)m-1≠(QP)m-1. The main result is the classification of all these algebras, implying that for each m⩾2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 435, no. 8, pp. 1823-1836, 2011]]>A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,…,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees, where in the case of characteristic different from two we ...

Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, vol. 434, no. 8, pp. 1920-1944, 2011]]>